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# 222m2 - [1(b at least one person receives his or her own...

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MATH 222 [A01] Fall 2011 Second Midterm Test Instructor: Dr. Jing Huang November 17, 2011 3:30 - 4:20pm Instructions: This question paper has fve pages plus cover. For each question, write out your solution/answer care±ully in the space provided. Use the back o± pages i± necessary. Marks will be deducted ±or incomplete or poorly presented solutions/answers. The Sharp EL-510R calculator is the only calculator permitted. No other aids (such as books, notes, ±ormula lists, or scratch paper) are allowed. Questions Value Marks 1 5 2 3 3 6 4 4 5 5 Total 23 Name: Id: .../1

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1 1. In how many ways can the nine letters M, A, T, H, I, S, F, U, N be permuted so that none of the words MAT H , IS and F UN occurs as consecutive letters? [5]
2 2. Twenty people check their hats at a theater. Use d n , which is the number of de- rangements of 1 , 2 , . . . , n , to answer the following questions: In how many ways can their hats be returned so that (a) no one receives his or her own hat?

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Unformatted text preview: [1] (b) at least one person receives his or her own hat? [1] (c) exactly one person receives his or her own hat? [1] 3 3. (a) Find a closed form (a.k.a. compact form) of the generating function for the sequence a = 0 and a k = (-2) k-1 , k ≥ 1 (i.e., 0 , 1 ,-2 , 4 ,-8 , 16 ,-32 , 64 , . . . ) [3] (b) Find the sequence represented by the generating function f ( x ) = x 2 / (1 + x ) 3 . [3] 4 4. Find the generating function for the sequence a n , n ≥ 0, where a n is the number of partitions of n into summands such that (a) each summand is an even number; [2] (b) each summand occurs an odd number of times. [2] 5 5. Find the generating function in a closed form which generates the sequence de±ned by the following recurrence relation (you do not need to solve for it). a n +1-2 a n = 5 , n ≥ 0; a = 1 . [5]-End-...
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222m2 - [1(b at least one person receives his or her own...

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